Abstract

We explain a proof of the Broué - Malle - Rouquier conjecture on Hecke
algebras of complex reflection groups, stating that the Hecke algebra of a finite
complex reflection group $ W$ is free of rank $ |W|$ over the algebra
of parameters, over a field of characteristic zero. This is based on previous work
of Losev, Marin - Pfeiffer, and Rains and the author.

Keywords

The goal of this note is to explain a proof of the Broué -
Malle - Rouquier conjecture ([Broué et al.1998], p. 178), stating that the
Hecke algebra of a finite complex reflection group $ W$ is free of rank
$ |W|$ over the algebra of parameters, over a field of characteristic zero.
This result is not original - it follows immediately from the results of [Losev2015], [Marin and Pfeiffer2017], and [Etingof and Rains2006], but it does not seem to
have been stated explicitly in the literature, so we state and prove it for future
reference.We note that there have been a lot of results on this conjecture for
particular complex reflection groups, reviewed in [Marin2015], e.g. [Ariki1995], [Ariki and Koike1994], [Marin2012], [Marin2014]; we are not giving the full list of
references here.

1. The Main Result

Let $ V$ be a finite dimensional complex vector space, and
$ W\subset GL(V)$ a finite complex reflection group, i.e., $ W$ is generated by
complex reflections (elements $ s$ such that $ {\rm rank}(1-s)=1$). Let
$ S\subset W$ be the set of reflections, and $ V_{\rm reg}:=V{\setminus} \cup_{s\in S}V^s$. Then by Steinberg's
theorem, $ W$ acts freely on $ V_{\rm reg}$. Let $ x\in V_{\rm reg}/W$ be a base
point. The braid group
of $ W$ is the group $ B_W:=\pi_1(V_{\rm {\bf }reg}/W,x)$. We have a surjective
homomorphism $ \pi: B_W\to W$ (corresponding to gluing back the reflection
hyperplanes $ V^s$), and $ {\rm Ker}\pi$ is called the pure braid group
of $ W$, denoted by $ PB_W$. For each $ s\in S$, let
$ T_s\in B_W$ be a path homotopic to a small circle around $ V^s$ (it is
defined uniquely up to conjugation). Also let $ n_s$ be the order of
$ s$. Then $ T_s^{n_s}\in PB_W$, and by the Seifert - van Kampen theorem,
$ PB_W$ is the normal closure of the subgroup of $ B_W$ generated
by $ T_s^{n_s}$, $ s\in S$. In other words, $ W$ is the quotient of
$ B_W$ by the relations $ T_s^{n_s}=1$, $ s\in S$.

Conjecture 1.2.

This conjecture is currently known for all irreducible complex
reflection groups except $ G_{17},\ldots,G_{21}$ (according to the Shephard - Todd
classification), and there is a hope that these cases can be proved as well using
a sufficiently powerful computer (see [Chavli2016a, Chavli2016b, Marin2015] for more
details). Also, it is shown in [Broué et al.1998] that to prove the
conjecture, it suffices to show that $ H(W)$ is spanned by $ |W|$
elements.

Our main result is

Theorem 1.3.

If $ K$ is a field of characteristic zero
then $ K\otimes_{\mathbb Z} H(W)$ is a free module over $ K\otimes_{\mathbb Z} R$ of rank $ |W|$. In
particular, if $ q: R\to K$ is a homomorphism, then the specialization
$ H_q(W):=K\otimes_R H(W)$ is a $ |W|$-dimensional $ K$-algebra.

Remark 1.4.

Theorem 1.3 is useful in many situations, for instance in the
representation theory of rational Cherednik algebras, where a number of
results were conditional on its validity for $ W$; see e.g. [Ginzburg et al.2003], 5.4, or [Shan2011], Section 2. Also, Theorem 1.3 implies a positive
answer to a question by Deligne and Mostow ([Deligne and Mostow1993], (17.20), Question 3),
which served as one of the motivations in [Broué et al.1998] (see [Broué et al.1998], p. 127).

2.
Proof of Theorem 1.3

First assume that $ K={\mathbb{C}}$. It also suffices to assume that
$ W$ is irreducible. In this case, possible groups $ W$ are
classified by Shephard and Todd ([Shephard1954]). Namely, $ W$ belongs to
an infinite series, or $ W$ is one of the exceptional groups
$ G_n$, $ 4\le n\le 37$. Among these, $ G_n$ with $ 4\le n\le 22$ are
rank $ 2$ groups, while $ G_n$ for $ n\ge 23$ are of rank
$ \ge 3$.

The case of the infinite series of groups is well known, see [Ariki1995], [Ariki and Koike1994], [Broué et al.1998]. So it suffices to focus on
the exceptional groups. Among these, the result is well known for Coxeter
groups, which are $ G_{23}=H_3$, $ G_{28}=F_4$, $ G_{30}=H_4$, $ G_{35}=E_6$,
$ G_{36}=E_7$, $ G_{37}=E_8$.

For the groups $ G_n$ for $ n=24,25,26,27,29,31,32,33,34$, the result was
established in [Marin and Pfeiffer2017] and references therein, see
[Marin2015], Subsection 4.1. Thus, Theorem 1.3 is known (in fact,
over any coefficient ring) for all $ W$ except those of rank
$ 2$.

In the rank 2 case, the following weak version of Theorem 1.3 was established.

Theorem 1.3 now follows from Theorem 2.1 and the following
theorem due to I. Losev.

Theorem 2.2.

([Losev2015], Theorem 1.1) For any $ W$
and any $ q: R\to {\mathbb{C}}$, there is a minimal two-sided ideal $ I\subset H_q(W)$ such
that $ H_q(W)/I$ is finite dimensional. Moreover, we have $ {\rm dim}H_q(W)/I=|W|$.

Namely, Theorems 2.1 and 2.2 imply that for any
character $ q: R\to {\mathbb{C}}$, the specialization $ H_q(W)$ has dimension
$ |W|$. This implies that for $ K={\mathbb{C}}$ the algebra $ K\otimes_{\mathbb Z}H(W)$ is a
projective $ K\otimes_{\mathbb Z}R$-module of rank $ |W|$ ([Hartshorne1977], Exercise 2.5.8(c)). Hence the
same is true for any field $ K$ of characteristic zero. But by Swan's
theorem ([Lam2006], Corollary 4.10), any finitely generated
projective module over the algebra of Laurent polynomials over a field is free.
Hence, the algebra $ K\otimes_{\mathbb Z}H(W)$ is a free $ K\otimes_{\mathbb Z}R$-module of rank
$ |W|$ (cf. also [Marin2014], Proposition 2.5). This proves Theorem
1.3.

Corollary 1.

Let $ K= {\mathbb{Z}}[1/N]$ for $ N\gg 0$. Then
$ K\otimes_{\mathbb{Z}}H(W)$ is a free $ K\otimes_{\mathbb Z}R$-module of rank $ |W|$. Hence the
same holds when $ K$ is a field of sufficiently large positive
characteristic.

Proof.

Theorem 2.1
is valid (with the same proof) over any coefficient ring (see e.g. [Marin2014], Theorem 2.14), i.e., for any
$ W$, the algebra $ H(W)$ is module-finite over $ R$.
Hence by Grothendieck's Generic Freeness Lemma ([Eisenbud1994], Theorem 14.4), there exists an
integer $ L>0$ such that $ H(W)[1/L]$ is a free $ \mathbb{Z}[1/L]$-module.

Now let $ v_1,\ldots,v_r$ be generators of $ H(W)$ over
$ R$, and $ e_i,\ldots,e_{|W|}\in H(R)$ be elements defining a basis of
$ \mathbb{Q}\otimes_{\mathbb Z}H(W)$ over $ \mathbb{Q} \otimes_{\mathbb{Z}}R$ (they exist by Theorem 1.3). Then
$ v_i=\sum_j a_{ij}e_j$ for some $ a_{ij}\in {\mathbb{Q}}\otimes_{\mathbb Z}R$. So for some integer $ D>0$ we
have $ Dv_i=\sum_j b_{ij}e_j$, with $ b_{ij}\in R$. Since $ H(W)[1/L]$ is a free
$ \mathbb{Z}[1/L]$-module, the same relation holds in $ H(W)[1/L]$. Thus, for
$ N=LD$, $ H(W)[1/N]$ is a free $ R[1/N]$-module with basis
$ e_1,\ldots,e_{|W|}$. ⬜

Remark 2.3.

1. The proof of
Theorem 1.3
does not extend to positive characteristic, since the proof of Theorem 2.2 uses complex
analysis (the Riemann - Hilbert correspondence).

2. The last step
of the proof of Theorem 1.3 (Swan's theorem) is really needed for purely
aesthetic purposes, to establish the original formulation of the conjecture on
the nose. As usual, for practical purposes it is normally sufficient to know only
that the algebra $ K\otimes_{\mathbb Z}H(W)$ is a projective
$ K\otimes_{\mathbb Z}R$-module. In fact, for most applications, including the ones
mentioned in Remark 1.4, already Losev's Theorem 2.2 is sufficient.

3. One would
like to have a stronger version of Theorem 1.3, giving a set-theoretical splitting $ W\to B_W$
of the homomorphism $ \pi$ whose image is a basis of the Hecke
algebra. For instance, when $ W$ is a Coxeter group, then such a
splitting is well known and is obtained by taking reduced expressions in the
braid group. Such a version is currently available (over any base ring) for all
irreducible complex reflection groups except $ G_{17},\ldots,G_{21}$, see [Marin2015], [Chavli2016a], [Chavli2016b].

4. Here is an
outline of the proof of Theorem 2.2 given in [Losev2015]. Let $ q=e^{2\pi ic}$, and let
$ \boldsymbol{H}_c(W)$ be the rational Cherednik algebra of $ W$ with
parameter $ c$, [Ginzburg et al.2003]. Let $ M\in {\mathcal O}_c(W)$ be a
module from the category $ \mathcal O$ for this algebra. It is shown in [Ginzburg et al.2003] that the localization of
$ M$ to the set $ \mathfrak{h}^{\rm reg}$ of regular points of the reflection
representation $ \mathfrak{h}$ of $ W$ is a vector bundle on
$ \mathfrak{h}^{\rm reg}$ with a flat connection. So for every $ x\in \mathfrak{h}^{\rm reg}$ we get a
monodromy representation of the braid group $ \pi_1(\mathfrak{h}^{\rm reg}/W)$ on the fiber
$ M_x$, which is shown in [Ginzburg et al.2003] to factor through
$ H_q(W)$. This representation is denoted by $ KZ(M)$, and the
functor $ M\mapsto KZ(M)$ is called the Knizhnik - Zamolodchikov (KZ) functor. It
is shown in [Ginzburg et al.2003] that the representation
$ KZ(M)$ of $ H_q(W)$ factors through a certain quotient
$ H_q'(W)$ of $ H_q(W)$ of dimension $ |W|$. Thus, Theorem
2.2 is
equivalent to the statement that every finite dimensional representation of
$ H_q(W)$ is of the form $ KZ(M)$ for some $ M$.

To show this, let $ \mathfrak{h}^{\rm sr}$ be the complement of the
intersections of pairs of distinct reflection hyperplanes in $ \mathfrak{h}$. Take
a finite dimensional representation $ V$ of $ H_q(W)$, and let
$ N=N_V$ be the vector bundle with a flat connection with regular
singularities on $ \mathfrak{h}^{\rm reg}$ corresponding to $ V$ under Deligne's
multidimensional Riemann - Hilbert correspondence. One then extends
$ N$ to a vector bundle $ \widetilde N$ on $ \mathfrak{h}^{\rm sr}$ compatibly
with the $ \boldsymbol{H}_c(W)$-action. One then defines $ M:=\Gamma(\mathfrak{h}^{\rm sr},\widetilde N)$ and shows that
$ M\in {\mathcal O}_c(W)$ and $ KZ(M)=V$, as desired.

5. Here is an
outline of the proof of Theorem 2.1 given in [Etingof and Rains2006]. For the infinite series of
complex reflection groups the result was proved in [Broué et al.1998]. Thus, let $ W\subset GL_2(\mathbb C)$
be an exceptional complex reflection group of rank $ 2$, of type
$ G_4,\ldots,G_{22}$. Then the intersection of $ W$ with the scalars is a
finite cyclic group generated by an element $ Z$. This element
defines a central element of $ H_q(W)$, which we will also call
$ Z$. Let $ W/\langle Z\rangle=G\subset PGL_2(\mathbb C)=SO_3(\mathbb C)$. Then $ G$ is the group of even
elements in a Coxeter group of type $ A_3$, $ B_3$, or
$ H_3$. Using the theory of length in these Coxeter groups, it is shown
that $ \mathbb{C} \otimes_{\mathbb Z}H(W)$ is generated by $ |G|$ elements as a module over
$ \mathbb{C} \otimes_{\mathbb{Z}}R[Z,Z^{-1}]$. Moreover, taking the determinant of the braid relation of this
algebra in its finite dimensional representations, we find that $ Z^d$ is
an element of $ \mathbb{C}\otimes_{\mathbb Z}R$ for some $ d$. This implies that
$ \mathbb{C}\otimes_{\mathbb Z}H(W)$ is a finite rank module over $ \mathbb{C}\otimes_{\mathbb Z}R$, as desired.

We note that this argument works over an arbitrary base ring. A
much more detailed description of this argument is given in [Chavli2016b].